What is Taiwan's other name?

It is formally known as the Republic of China. Taiwan was named Formosa ("beautiful [island]") by Portuguese explorers, so Westerners did know the island as "Formosa" until the 20th century.

On mainland China, Taiwan is written 台湾, and spelled Táiwan in pinyin. On Taiwan, the name is written 臺灣, and spelled Taiwan or T'ai-wan in roman letters. The difference in Chinese characters has to do with the fact that on mainland China they now use "simplified characters" that the Taiwanese do not use.

When the Dutch were running a colonial trade company based on Taiwan, they also used the spelling Tayoan, which is pronounced the same as "Taiwan."